A proposal for the algebra of a novel noncommutative spacetime

TL;DR

This paper introduces a Lorentz-invariant noncommutative spacetime algebra that models quantum gravitational effects, revealing minimal length scales and a fuzzy causal structure that converges to classical spacetime in the limit.

Contribution

It constructs a novel quantum spacetime algebra using noncommutative geometry and spectral theory, preserving Lorentz symmetry and incorporating quantum gravitational effects.

Findings

Quantum spacetime exhibits minimal length effects.

Noncommutativity induces a fuzzy causal structure.

Classical light cone is recovered in the limit.

Abstract

We investigate the quantum structure of spacetime at fundamental scales via a novel, Lorentz-invariant noncommutative coordinate framework. Building on insights from noncommutative geometry, spectral theory, and algebraic quantum field theory, we systematically construct a quantum spacetime algebra whose geometric and causal properties are derived from first principles. Using the Weyl algebra formalism and the Gelfand--Naimark--Segal (GNS) construction, we rigorously define operator-valued coordinates that respect Lorentz symmetry and encode quantum gravitational effects through nontrivial commutation relations. We show how the emergent quantum spacetime exhibits minimal length effects, which deliver both classical Minkowski distances and quantum corrections proportional to the Planck length squared. Furthermore, we establish that noncommutativity respects a fuzzy form of causality, where the quantum causal structure gives back the light cone in the classical limit, vanishing for spacelike separations and encoding a time orientation for timelike intervals.